HENRI POINCARE NORDMANN. 747 



coveries were numerous and would have sufficed for the glory of 

 several mathematicians. 



It was especially in the theory of differential equations that the 

 genius of Poincare showed itself. If he spent on them the greater 

 part of his intellectual resources it was without doubt because most 

 of the problems offered in the physical study of the universe led to 

 just such equations. Newton was the first to show that the state of 

 a moving system, or, more generally, that of the universe, depends 

 only on its immediately preceding state, and that all the changes in 

 nature take place in a continuous manner. True, the ancients in 

 their adage, ''Natura non fecit saltus," had an inkling of it. But 

 Newton was the fu'st, with the great pliilosophers of the seventeenth 

 century, to free the idea from the scholastic errors which perverted 

 it and then to assure its development. A law, then, is only the neces- 

 sary relation between the present state of the world and that imme- 

 diately preceding. It is a consequence of this that in place of study- 

 ing du'ectly a succession of events we may limit ourselves to consid- 

 ermg the manner in which two successive phenomena occur ; in other 

 words, we may express om' succession by a differential equation. 

 All natural laws which have been discovered are only differential 

 equations. Looking at it slightly differently, such equations have 

 been possible in physics because the greater part of physical phe- 

 nomena may be analyzed as the succession of a great number of ele- 

 mentary events, 'infinitesimals," all similar. 



The knowledge of this elementary fact allows us to construct the 

 differential equation and we have then to use only a method of summa- 

 tion in order to deduce an observable and verifiable complex phe- 

 nomenon. This mathematical operation of summation is called the 

 "integration" of the differential equation. In the greater number 

 of cases this integration is impossible, and perhaps all progress in 

 physics depends on perfecting the process of integration. That was 

 the principal work of Poincare in mathematics. And in that line 

 his work was amazing, especially in the development of those now 

 famous functions, the simplest of which are known as the Fuchsian 

 functions (named after the German mathematician Fuchs, whose 

 work had been of aid to Pouicar^). We may represent by these new 

 transcendental functions, which are also called automorphic, curves 

 of any degree and solve all linear differential equations with algebraic 

 coefficients. Poincar§ thus gave us, using the apt expression of his 

 colleague, M. Humbert, of the Academie des sciences, "the keys of 

 the algebraic world." Poincare himself used these algebraic tools 

 in his researches in celestial mechanics. 



To tell the truth, the Newtonian idea as to the continuity of phys- 

 ical phenomena has of late been somewhat battered down in several 



